From Newton to Navier–Stokes, or how to connect fluid mechanics equations from microscopic to macroscopic scales
HTML articles powered by AMS MathViewer
- by Isabelle Gallagher PDF
- Bull. Amer. Math. Soc. 56 (2019), 65-85 Request permission
Abstract:
In this survey we present an overview of some mathematical results concerning the passage from the microscopic description of fluids via Newton’s laws to the macroscopic description via the Navier–Stokes equations.References
- Roger Keith Alexander, THE INFINITE HARD-SPHERE SYSTEM, ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of California, Berkeley. MR 2625918
- Roger Alexander, Time evolution for infinitely many hard spheres, Comm. Math. Phys. 49 (1976), no. 3, 217–232. MR 434284, DOI 10.1007/BF01608728
- J. d’Alembert, Essai d’une nouvelle théorie de la résistance des fluides (1752), Paris.
- Nathalie Ayi, From Newton’s law to the linear Boltzmann equation without cut-off, Comm. Math. Phys. 350 (2017), no. 3, 1219–1274. MR 3607474, DOI 10.1007/s00220-016-2821-6
- Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550, DOI 10.1007/978-3-642-16830-7
- C. Bardos, F. Golse, and C.D. Levermore, Fluid dynamic limits of the Boltzmann equation I, J. Stat, Phys. 63 (1991), 323–344.
- Claude Bardos, François Golse, and C. David Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (1993), no. 5, 667–753. MR 1213991, DOI 10.1002/cpa.3160460503
- C. Bardos, R. Santos, and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc. 284 (1984), no. 2, 617–649. MR 743736, DOI 10.1090/S0002-9947-1984-0743736-0
- H. van Beijeren, O. E. Lanford III, J. L. Lebowitz, and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Statist. Phys. 22 (1980), no. 2, 237–257. MR 560556, DOI 10.1007/BF01008050
- Thierry Bodineau, Isabelle Gallagher, and Laure Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math. 203 (2016), no. 2, 493–553. MR 3455156, DOI 10.1007/s00222-015-0593-9
- Thierry Bodineau, Isabelle Gallagher, and Laure Saint-Raymond, From hard sphere dynamics to the Stokes-Fourier equations: an $L^2$ analysis of the Boltzmann-Grad limit, Ann. PDE 3 (2017), no. 1, Paper No. 2, 118. MR 3625187, DOI 10.1007/s40818-016-0018-0
- Thierry Bodineau, Isabelle Gallagher, and Laure Saint-Raymond, Derivation of an Ornstein-Uhlenbeck process for a massive particle in a rarified gas of particles, Ann. Henri Poincaré 19 (2018), no. 6, 1647–1709. MR 3806440, DOI 10.1007/s00023-018-0674-6
- N. N. Bogoliubov, Problems of a dynamical theory in statistical physics, Studies in Statistical Mechanics, Vol. I, North-Holland, Amsterdam; Interscience, New York, 1962, pp. 1–118. MR 0136381
- L. Boltzmann, Weitere Studien uber das Warmegleichgenicht unfer Gasmolakular, Sitzungsberichte der Akademie der Wissenschaften 66 (1872), 275–370. Translation : Further studies on the thermal equilibrium of gas molecules, in Kinetic Theory 2, 88–174, Ed. S.G. Brush, Pergamon, Oxford (1966).
- L. Boltzmann, Leçons sur la théorie des gaz, Gauthier-Villars (Paris, 1902-1905). Ré-édition Jacques Gabay, 1987.
- M. Born and H. S. Green, A general kinetic theory of liquids. I. The molecular distribution functions, Proc. Roy. Soc. London Ser. A 188 (1946), 10–18. MR 23769, DOI 10.1098/rspa.1946.0093
- L. A. Bunimovich and Ya. G. Sinaĭ, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/81), no. 4, 479–497. MR 606459, DOI 10.1007/BF02046760
- Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys. 74 (1980), no. 1, 71–95. MR 575897, DOI 10.1007/BF01197579
- Marco Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana 13 (1997), no. 3, 515–541. MR 1617394, DOI 10.4171/RMI/229
- Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028, DOI 10.1007/978-1-4612-1039-9
- C. Cercignani, V. I. Gerasimenko, and D. Ya. Petrina, Many-particle dynamics and kinetic equations, Mathematics and its Applications, vol. 420, Kluwer Academic Publishers Group, Dordrecht, 1997. Translated from the Russian manuscript by K. Petrina and V. Gredzhuk. MR 1472233, DOI 10.1007/978-94-011-5558-8
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620, DOI 10.1007/978-1-4419-8524-8
- Sydney Chapman and T. G. Cowling, The mathematical theory of non-uniform gases: An account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, New York, 1960. MR 0116537
- A. De Masi, R. Esposito, and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42 (1989), no. 8, 1189–1214. MR 1029125, DOI 10.1002/cpa.3160420810
- Ryan Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 885–952. MR 3803778, DOI 10.1007/s00205-018-1229-1
- L. Desvillettes and F. Golse, A remark concerning the Chapman-Enskog asymptotics, Advances in kinetic theory and computing, Ser. Adv. Math. Appl. Sci., vol. 22, World Sci. Publ., River Edge, NJ, 1994, pp. 191–203. MR 1323184
- L. Desvillettes and M. Pulvirenti, The linear Boltzmann equation for long-range forces: a derivation from particle systems, Math. Models Methods Appl. Sci. 9 (1999), no. 8, 1123–1145. MR 1722064, DOI 10.1142/S0218202599000506
- L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions, J. Statist. Phys. 104 (2001), no. 5-6, 1173–1189. MR 1859001, DOI 10.1023/A:1010461929872
- R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321–366. MR 1014927, DOI 10.2307/1971423
- Matthew Dobson, Frédéric Legoll, Tony Lelièvre, and Gabriel Stoltz, Derivation of Langevin dynamics in a nonzero background flow field, ESAIM Math. Model. Numer. Anal. 47 (2013), no. 6, 1583–1626. MR 3110489, DOI 10.1051/m2an/2013077
- D. Dürr, S. Goldstein, and J. L. Lebowitz, A mechanical model of Brownian motion, Comm. Math. Phys. 78 (1980/81), no. 4, 507–530. MR 606461, DOI 10.1007/BF02046762
- D. Dürr, S. Goldstein, and J. L. Lebowitz, A mechanical model for the Brownian motion of a convex body, Z. Wahrsch. Verw. Gebiete 62 (1983), no. 4, 427–448. MR 690569, DOI 10.1007/BF00534196
- R. Esposito, R. Marra, and H. T. Yau, Navier-Stokes equations for stochastic particle systems on the lattice, Comm. Math. Phys. 182 (1996), no. 2, 395–456. MR 1447299, DOI 10.1007/BF02517896
- Leonhard Euler, Principles of the motion of fluids, Phys. D 237 (2008), no. 14-17, 1840–1854. English adaptation by Walter Pauls. MR 2449769, DOI 10.1016/j.physd.2008.04.019
- Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, DOI 10.1007/BF00276188
- Isabelle Gallagher, Laure Saint-Raymond, and Benjamin Texier, From Newton to Boltzmann: hard spheres and short-range potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013. MR 3157048
- Giovanni Gallavotti, Statistical mechanics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1999. A short treatise. MR 1707309, DOI 10.1007/978-3-662-03952-6
- François Golse, The mean-field limit for the dynamics of large particle systems, Journées “Équations aux Dérivées Partielles”, Univ. Nantes, Nantes, 2003, pp. Exp. No. IX, 47. MR 2050595
- François Golse and Laure Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math. 155 (2004), no. 1, 81–161. MR 2025302, DOI 10.1007/s00222-003-0316-5
- François Golse and Laure Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9) 91 (2009), no. 5, 508–552 (English, with English and French summaries). MR 2517786, DOI 10.1016/j.matpur.2009.01.013
- A. N. Gorban, Hilbert’s sixth problem: the endless road to rigour, Philos. Trans. Roy. Soc. A 376 (2018), no. 2118, 20170238, 10. MR 3797486, DOI 10.1098/rsta.2017.0238
- Alexander N. Gorban and Ilya Karlin, Hilbert’s 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 2, 187–246. MR 3166040, DOI 10.1090/S0273-0979-2013-01439-3
- Harold Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331–407. MR 33674, DOI 10.1002/cpa.3160020403
- David Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912), no. 4, 562–577 (German). MR 1511713, DOI 10.1007/BF01456676
- Richard Holley, The motion of a heavy particle in an infinite one dimensional gas of hard spheres, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 181–219. MR 283907, DOI 10.1007/BF00536757
- Reinhard Illner and Mario Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Comm. Math. Phys. 105 (1986), no. 2, 189–203. MR 849204, DOI 10.1007/BF01211098
- R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: “Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum” [Comm. Math. Phys. 105 (1986), no. 2, 189–203; MR0849204 (88d:82061)] and “Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum” [ibid. 113 (1987), no. 1, 79–85; MR0918406 (89b:82052)] by Pulvirenti, Comm. Math. Phys. 121 (1989), no. 1, 143–146. MR 985619
- Tosio Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\textbf {R}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, DOI 10.1007/BF01174182
- J. G. Kirkwood, The statistical mechanical theory of transport processes I. General theory, Journal of Chemical Physics 14 (1946), 180–202.
- Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843, DOI 10.1006/aima.2000.1937
- M. Lachowicz, On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci. 9 (1987), no. 3, 342–366. MR 908596, DOI 10.1002/mma.1670090127
- Oscar E. Lanford III, Time evolution of large classical systems, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 1–111. MR 0479206
- J. L. Lebowitz and H. Spohn, Steady state self-diffusion at low density, J. Statist. Phys. 29 (1982), no. 1, 39–55. MR 676928, DOI 10.1007/BF01008247
- P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1938147, DOI 10.1201/9781420035674
- Pierre Gilles Lemarié-Rieusset, The Navier-Stokes problem in the 21st century, CRC Press, Boca Raton, FL, 2016. MR 3469428, DOI 10.1201/b19556
- Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
- J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, Journal de Mathématiques Pures et Appliquées, 12 (1933), 1–82.
- P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173–193, 195–211. MR 1842343, DOI 10.1007/s002050100143
- Nader Masmoudi and Laure Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math. 56 (2003), no. 9, 1263–1293. MR 1980855, DOI 10.1002/cpa.10095
- Karsten Matthies and Florian Theil, A semigroup approach to the justification of kinetic theory, SIAM J. Math. Anal. 44 (2012), no. 6, 4345–4379. MR 3028562, DOI 10.1137/120865598
- C. Navier, Mémoire sur les lois du mouvement des fluides, Mémoire de l’Académie des Sciences de l’Institut de France, 6 (1822), 375–394.
- L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differential Geometry 6 (1972), 561–576. MR 322321, DOI 10.4310/jdg/1214430643
- Takaaki Nishida, A note on a theorem of Nirenberg, J. Differential Geometry 12 (1977), no. 4, 629–633 (1978). MR 512931
- S. Olla, S. R. S. Varadhan, and H.-T. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise, Comm. Math. Phys. 155 (1993), no. 3, 523–560. MR 1231642, DOI 10.1007/BF02096727
- Fabrice Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in $\textbf {R}^3$, Rev. Mat. Iberoamericana 14 (1998), no. 1, 71–93. MR 1639283, DOI 10.4171/RMI/235
- M. Pulvirenti, C. Saffirio, and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys. 26 (2014), no. 2, 1450001, 64. MR 3190204, DOI 10.1142/S0129055X14500019
- J. Quastel and H.-T. Yau, Lattice gases, large deviations, and the incompressible Navier-Stokes equations, Ann. of Math. (2) 148 (1998), no. 1, 51–108. MR 1652971, DOI 10.2307/120992
- Laure Saint-Raymond, From the BGK model to the Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 271–317 (English, with English and French summaries). MR 1980313, DOI 10.1016/S0012-9593(03)00010-7
- Laure Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, vol. 1971, Springer-Verlag, Berlin, 2009. MR 2683475, DOI 10.1007/978-3-540-92847-8
- M. Slemrod, From Boltzmann to Euler: Hilbert’s 6th problem revisited, Comput. Math. Appl. 65 (2013), no. 10, 1497–1501. MR 3061719, DOI 10.1016/j.camwa.2012.08.016
- M. Slemrod, The problem with Hilbert’s 6th problem, Math. Model. Nat. Phenom. 10 (2015), no. 3, 6–15. MR 3371918, DOI 10.1051/mmnp/201510302
- Marshall Slemrod, Hilbert’s sixth problem and the failure of the Boltzmann to Euler limit, Philos. Trans. Roy. Soc. A 376 (2018), no. 2118, 20170222, 12. MR 3797494, DOI 10.1098/rsta.2017.0222
- Herbert Spohn, Boltzmann hierarchy and Boltzmann equation, Kinetic theories and the Boltzmann equation (Montecatini, 1981) Lecture Notes in Math., vol. 1048, Springer, Berlin, 1984, pp. 207–220. MR 740726, DOI 10.1007/BFb0071883
- H. Spohn, Large scale dynamics of interacting particles, Texts and Monographs in Physics, Springer-Verlag, Berlin/Heidelberg, 1991, 174 pp.
- G. Stokes, On the theories of internal friction of fluids in motion and of the equilibrium and motion of elastic solids, Trans. Camb. Phil. Soc. 8 (1845), 287–319.
- Seiji Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad. 50 (1974), 179–184. MR 363332
- Seiji Ukai, Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demi-espace, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 6, Ai, A317–A320 (French, with English summary). MR 445138
- Seiji Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Japan J. Indust. Appl. Math. 18 (2001), no. 2, 383–392. Recent topics in mathematics moving toward science and engineering. MR 1842918, DOI 10.1007/BF03168581
- Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. MR 1942465, DOI 10.1016/S1874-5792(02)80004-0
- J. Yvon, La théorie statistique des fluides et l’équation d’état, Actual. Sci. et Indust. 203 (Paris, Hermann), 1935.
Additional Information
- Isabelle Gallagher
- Affiliation: Université Paris-Diderot, Sorbonne Paris Cité, France
- Address at time of publication: DMA, École Normale Suptérieure de Paris, UMR 8553, France
- MR Author ID: 617258
- Email: gallagher@math.ens.fr
- Received by editor(s): June 11, 2018
- Published electronically: September 18, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 56 (2019), 65-85
- MSC (2010): Primary 76D05, 82B40, 82C22
- DOI: https://doi.org/10.1090/bull/1650
- MathSciNet review: 3886144